1. No category

BioMedical Engineering OnLine
This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted
PDF and full text (HTML) versions will be made available soon.
Dynamic behavior of suture-anastomosed arteries and implications to vascular
surgery operations
BioMedical Engineering OnLine 2015, 14:1
doi:10.1186/1475-925X-14-1
Panayiotis C Roussis ([email protected])
Antonios E Giannakopoulos ([email protected])
Haralambia P Charalambous ([email protected])
Demetra C Demetriou ([email protected])
Georgios P Georghiou ([email protected])
ISSN
Article type
1475-925X
Research
Submission date
9 July 2014
Acceptance date
22 December 2014
Publication date
6 January 2015
Article URL
http://www.biomedical-engineering-online.com/content/14/1/1
This peer-reviewed article can be downloaded, printed and distributed freely for any purposes (see
copyright notice below).
Articles in BioMedical Engineering OnLine are listed in PubMed and archived at PubMed Central.
For information about publishing your research in BioMedical Engineering OnLine or any BioMed
Central journal, go to
http://www.biomedical-engineering-online.com/authors/instructions/
For information about other BioMed Central publications go to
http://www.biomedcentral.com/
© 2015 Roussis et al.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain
Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Dynamic behavior of suture-anastomosed arteries
and implications to vascular surgery operations
Panayiotis C Roussis1*
*
Corresponding author
Email: [email protected]
Antonios E Giannakopoulos2
Email: [email protected]
Haralambia P Charalambous1
Email: [email protected]
Demetra C Demetriou1
Email: [email protected]
Georgios P Georghiou3,4
Email: [email protected]
1
Department of Civil & Environmental Engineering, University of Cyprus,
Nicosia CY-1678, Cyprus
2
Department of Civil Engineering, University of Thessaly, Volos GR-38334,
Greece
3
Cardiothoracic Surgery Department, American Medical Center, Nicosia CY1311, Cyprus
4
Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv 69978, Israel
Abstract
Background
Routine vascular surgery operations involve stitching of disconnected human arteries with
themselves or with artificial grafts (arterial anastomosis). This study aims to extend current
knowledge and provide better-substantiated understanding of the mechanics of end-to-end
anastomosis through the development of an analytical model governing the dynamic behavior
of the anastomotic region of two initially separated arteries.
Methods
The formulation accounts for the arterial axial-circumferential deformation coupling and
suture-artery interaction. The proposed model captures the effects of the most important
parameters, including the geometric and mechanical properties of artery and sutures, number
of sutures, loading characteristics, longitudinal residual stresses, and suture pre-tensioning.
Results
Closed-form expressions are derived for the system response in terms of arterial radial
displacement, anastomotic gap, suture tensile force, and embedding stress due to suture-artery
contact interaction. Explicit objective functionalities are established to prevent failure at the
anastomotic interface.
Conclusions
The mathematical formulation reveals useful interrelations among the problem parameters,
thus making the proposed model a valuable tool for the optimal selection of materials and
improved functionality of the sutures. By virtue of their generality and directness of
application, the findings of this study can ultimately form the basis for the development of
vascular anastomosis guidelines pertaining to the prevention of post-surgery implications.
Keywords
End-to-end anastomosis, Anastomotic gap, Sutures, Arterial-wall stress, Suture stress, Failure
criteria
Background
Predictive medicine and therapeutic decision-making requires thorough understanding of the
human biological activities in order to develop a proper physical model and obtain the
optimal solution of the problem. Vascular surgery operations treat vascular diseases, trafficrelated and other serious injuries entailing violent artery damage. Among the vascular
disorders, atherosclerosis and aneurysms are the most frequently encountered. During a
typical arterial reconstruction, the diseased artery segment is removed, and the healthy
segments are stitched together, either directly or through the insertion of an artificial graft
(end-to-end anastomosis). In any case, the mechanical behavior of the anastomotic region is
in principle comparable, on account of the fact that modern grafts tend to exhibit similar
geometric and stiffness characteristics with those of arteries.
Several studies have examined the induced arterial-wall stresses in vascular anastomosis
models [1-7]. However, most of them limit their research to specific arterial geometries,
while others ignore the stress concentrations due to suture-artery interaction, or the important
axial-circumferential deformation coupling in the artery response. Moreover, most of the
published studies rely solely on finite element analyses rather than on analytical models of
end-to-end and end-to-side anastomosis.
In particular, Ballyk et al. [1] studied an end-to-end and an end-to-side anastomosis by use of
finite-element analysis, with the sutures modeled as discrete points along the suture line. The
expected stress concentration of the stitching area resulted in excessive values due to the
point-like modelling approach as such. Leuprecht et al. [2] and Perktold et al. [3] utilized
three end-to-side anastomosis models, each one concerning a different technique. Their
finite-element analysis yielded the wall shear stresses and maximum principal stresses for
each case. In particular, the latter study modeled the stitches in detail using three-dimensional
elements for the junction. In another finite element study, Cacho et al. [6] investigated the
effect of the insertion angle and incision length of coronary arterial bypass models, though
without modelling explicitly the response of individual stitches. More recently, Schiller et al.
[7] studied an end-to-end anastomosis by using a fluid–structure coupling algorithm, with the
sutures simulated as an anastomotic interface.
In addition to theoretical studies, which on one hand are limited and on the other hand require
numerical methods to calculate the response [8], several experimental studies have been
performed over the years (see for example [9-13]) to investigate the compliance of the
anastomotic region. Lyman et al. [9] found that vascular grafts should have compliance
approximately equal to that of the host artery, and that compliance of synthetic grafts may
decrease with time. Hasson et al. [10] investigated an end-to-end anastomosis between
isocompliant arterial grafts from dogs and found that a para-anastomotic hypercompliant
zone (PHZ), which promotes subintimal hyperplasia (SIH), exists near the suturing region.
The PHZ is also a zone of increased cyclic stretch. The compliance at this region increases up
to 50% compared to the compliance away from the stitching region. In a later study, Hasson
et al. [11] concluded that the suture technique affects significantly the compliance of the
anastomotic region. In particular, they showed that the PHZ phenomenon occurs more
frequently for anastomosis of the continuous stitching technique than the interrupted stitching
technique. In addition, they observed that increased longitudinal stress of the arterial vessel
reduce the compliance. This phenomenon can be justified from the fact that longitudinal prestress affects the mechanical properties of dog arteries [14]. Ulrich et al. [13] investigated an
end-to-end anastomosis between pig aortic grafts and found that PHZ does not exist for this
case. They also suggested that the main factor affecting the anastomotic response is the suture
line itself.
Evidently, little work has been published on the dynamic analysis of the stitched anastomotic
region. Indeed, related review articles clearly point out the lack of such analyses [15]. In
addressing this need, this paper proposes a comprehensive analytical model based on
structural analysis, aiming to provide a better understanding of the dynamic response of endto-end arterial anastomosis, with emphasis on the suture-artery interaction and the axialcircumferential deformation coupling in the artery response. This study seeks to extend
current knowledge and provide practical suggestions for the optimal selection of materials
and improved functionality of the sutures in vascular surgery operations.
Methods
This section presents the mathematical formulation governing the dynamic behavior of endto-end arterial anastomosis. The system response is described in terms of the radial
displacement of artery, the anastomotic gap, the suture tensile force, and the embedding stress
due to suture-artery contact interaction. Based on these response quantities, comprehensive
objective functionalities are established in order to prevent failure at the anastomotic
interface.
Arterial model
From the mechanics point of view, the human arterial system can be idealized as a system of
cylindrical elastic pipes that transport blood under pressure provided from the heart (dynamic
loading). The arterial tissue is heterogeneous, consisting of three inhomogeneous layers. Its
mechanical properties depend on the artery location, age, disease, and other physiological
states [16,17]. In general, the mechanical behavior of the arterial tissue does not obey
Hooke’s law [18,19], exhibiting anisotropic nonlinear behavior for finite deformations.
Moreover, the response of biological tissues is affected by the existence of residual stresses
[20]. In this study, the arterial wall is assumed homogeneous and its mechanical response
linear elastic. The assumed elastic properties in the model incorporate in an average sense the
tangential stiffness, the anisotropy, the inhomogeneity, and the residual stresses of the artery
walls.
In this study, the blood vessel is modeled as an elastic cylindrical pipe with wall thickness h
and radius R (Figure 1(a)). The mathematical formulation developed herein is based on the
following assumptions: (a) the arterial wall thickness is small compared to the radius of the
centerline of the ring therefore the radial stresses are not considered; (b) the centerline of the
ring in the undeformed state forms a full circle with radius R; (c) the cross-section is axially
symmetric and constant around the circle, implying that the arterial wall has constant
thickness; (d) no boundary constraints are applied on the ring; (e) the effects of rotary inertia
and shear deformation are neglected; (f) the arterial tissue consists of a single homogeneous
layer and behaves as a simple orthotropic linear elastic material (the mechanical properties in
the radial and circumferential directions − which are the same − differ from those in the
longitudinal direction, ignoring the Poisson effect in the orthotropy constitutive law); and (g)
viscous effects are ignored. The simplified orthotropic model adopted in this work utilizes
two elastic constants Eθ and EL which are the plane strain elastic moduli suggested by Schajer
et al. [21], in the circumferential and the longitudinal directions respectively.
Figure 1 Idealized arterial system. (a) Arterial model, (b) typical element of circular ring.
Response to general dynamic loading
Under the assumption of axial symmetry, the system undergoes in-plane extensional vibration
due to a uniformly-distributed wall pressure p(t). The differential equation governing the
radial displacement u(t) of the vibrating arterial ring can be derived by considering the
equilibrium of forces acting on an element of the circular ring (Figure 1(b)). The resulting
equation is (see Appendix A1)
ρh
d 2u ( t ) Eθ h
+ 2 u (t ) = p (t )
dt
R
(1)
where ρ denotes the density of the arterial tissue, h the wall thickness, R the radius of the
undeformed centerline of the ring, and Eθ the arterial Young’s modulus in the circumferential
direction.
The first term in equation (1) represents the radial inertia force acting on element abcd of the
circular ring, while the second term represents the circumferential tensile force developed on
the element cross section assuming linear-elastic behavior.
Equation (1) is identical to the classical second-order differential equation governing the
response of an undamped single-degree-of-freedom system to arbitrary force. The circular
frequency of the system is readily obtained as
ωn =
1
R
Eθ
(2)
ρ
Response to pulse-type loading
A normal cardiac cycle consists of two major functional periods: systolic and diastolic.
Figure 2(a) shows the aortic pressure–time profile as proposed by Zhong et al. [22]. The time
interval 0 ≤ t ≤ ts represents the aortic systolic phase, during which the arterial walls inflate
due to the maximum overstress pressure. The time interval ts < t ≤ tcp represents the aortic
diastolic phase.
Figure 2 Blood pressure time-profiles. (a) Typical aortic pressure time-profile following
Zhong et al. [22], (b) arterial pulse time-history approximation. (100 mmHg = 13.33kPa).
During a vascular surgery operation the blood flow is interrupted. The first loading cycle,
immediately after the flow is restored, is approximated in this study by the loading shown in
Figure 2(b). In this case, the internal pressure is abruptly increased from zero to the
maximum systolic pressure. The assumed loading is expressed mathematically as
, 0 ≤ t ≤ ts
 ps

p(t ) = 
ps − pd
p
−
( t − ts ) , ts < t ≤ tcp
s

t
−
t
cp
s

(3)
where ps is the maximum systolic pressure, pd the diastolic pressure, ts the systolic-phase
duration, and tcp the total duration of the cardiac pulse.
The radius R is measured at zero blood pressure and in vivo length, implying the artery is in
its pre-stressed state. During surgery, longitudinal residual stresses are released, forcing the
artery to decrease its length and increase its diameter. When subsequently the stitching takes
place, the arterial diameter and length return to their prior condition. The residual-stress
effect is taken into account as an initial displacement u(0) = u0 (and so the far-field stresses
can be inserted), equal to the difference of the increased radius (relieved from axial residual
stresses) and radius R, and initial velocity ˙ 0
0.
The total response of the system as a function of time is obtained as (see Appendix A2)

ps R 2
u
cos
ω
t
+
(1 − cos ωnt ) ,
n
 0
E
h
θ


u (t ) = 

2 

sin ωn ( t − t s ) 

 u0 cos ωn t + ps R  ps − pd t s − t +
 + ps (1 − cos ωn t )  ,
ωn
Eθ h  tcp − t s 



0 ≤ t ≤ ts
(4)
ts < t ≤ tcp
in which the first term (free-vibration response) is associated with the residual-stress effect
and the second term is associated with the response to the assumed pulse-type loading.
The static response of the system, which corresponds to the displacement caused when the
maximum pressure ps is applied statically, is identified as
ust = ps R 2 / Eθ h
(5)
Of particular interest is the maximum arterial displacement, which is associated with the
critical response of the anastomotic region. The maximum displacement occurs either in the
time interval 0 ≤ t ≤ ts or ts < t ≤ tcp, depending on the system natural period Tn = 2π/ωn and
the loading characteristics. The overall maximum displacement can be calculated from the
following expression (for |u0|/ust < 1): (see Appendix A3)
 2 p R 2
R 2  ps − pd
umax = max  s − u0 , u0 cos ωn t1 +

Eθ h  tcp − t s
 Eθ h
 

sin ωn ( t1 − t s ) 
ω
t
−
t
+
+
p
1
−
cos
t
(
)
 s 1

s
n 1 
ωn


 
where t1 is the time instant corresponding to the maximum response in the diastolic phase,
given by
t1 =
1
ωn
( cos
−1
A1 + tan −1 A2 )
(7)
in which
A1 =
A2 =
R 2 p s − pd
Eθ h tcp − ts
2
 ps R 2 R 2 ps − pd sin ωn t s
  R 2 ps − pd sin ωn ts
+
− u0  + 

 
ωn
ωn
 Eθ h Eθ h tcp − ts
  Eθ h tcp − t s
ps R 2 R 2 ps − pd sin ωn ts
+
− u0
ωn
Eθ h Eθ h tcp − ts
R 2 ps − pd sin ωn ts
ωn
Eθ h tcp − t s



2
(8)
(9)
Figure 3 plots the normalized maximum deformation umax/ust as a function of the ratio ts/Tn
for different values of initial displacement, and for typical values of diastolic pressure,
maximum systolic pressure, and cardiac pulse duration (pd = 80 mmHg, ps = 120 mmHg, tcp =
1 sec). In particular, Figure 3(a) plots the normalized response for a typical cardiac pulse with
fixed systolic-phase duration, ts = 0.35 sec, and varying system natural period Tn. As can be
seen in Figure 3(a), the response exhibits an ascending curved profile for low values of ts/Tn,
reaching practically a plateau for high values of ts/Tn. Τhe threshold value of ts/Tn that defines
the boundary between the ascending part and the plateau depends on the loading
characteristics. For the parameters used in Figure 3(a), the threshold value of ts/Tn is
approximately 0.4. Figure 3(b) plots the normalized response for fixed natural period Tn = 0.9
sec and varying systolic-phase duration ts. The high value of natural period (Tn = 0.9 sec)
implies severely damaged artery walls, with the artery elasticity modulus practically tending
to zero.
(6)
Figure 3 Normalized maximum displacement as a function of ts/Tn and the parameter
u0/ust. Plots for (a) ts = 0.35 sec, (b) Tn = 0.9 sec.
Anastomosis model
A schematic of the end-to-end anastomosis model considered in this study is shown in Figure
4(a). The proximal and distal artery segments are connected together with a total of Ns
stitches. Each artery segment has length L, radius R, and Young’s modulus in the longitudinal
direction EL. The stitches have radius rs, cross-sectional area As = πrs2, and Young’s modulus
Es. The suture material is legitimately considered to be linear elastic for elongations up to
20% [23]. The distance between stitching holes that are symmetrically located across the
separation plane is denoted by ls (with the assumption that 2L ≫ ls). Different stitching
patterns are considered, resulting in different suture loading. Figure 4(e) depicts the
interrupted stitching scheme, whereas Figure 4(f) depicts the continuous (running) stitching
scheme. The particular loading condition associated with each stitching scheme is accounted
for in the analysis by means of a participation factor a. The interrupted stitching scheme
corresponds to a maximum participation factor a = 2, whereas the continuous stitching
scheme (with diagonal at 45° angle), corresponds to participation factor a = 1.707. The
participation factor is derived from the local equilibrium of forces at the suture line that
passes without friction through the stitch hole. The participation factor indicates the
alignment of the stitches along the longitudinal direction (the remaining part (2 − a) indicates
that the system is in torsion with limited relevance to the present problem). Moreover, the
stitching holes and the suture are considered to have almost equal diameters. Therefore, the
suture segment penetrating the arterial wall is almost undeformable, due to friction forces
developed between the arterial wall and the suture. The model also considers the pretensioning of stitches [24], denoted herein by the force
. This is the force exerted by the
surgeon in tying the knot of the suture.
Figure 4 End-to-end anastomosis analysis between isocompliant blood vessels. (a)
Anastomosis model (at-rest state); the artery is clamped at the far ends and no pressure is
transmitted at this stage since the artery is emptied from the blood, (b) unrestrained deformed
state of artery (without sutures); the blood volume is conserved, (c) deformed state of
anastomotic region due to dynamic loading, (d) forces acting on end-element of artery
segment, (e) interrupted stitching scheme, (f) continuous stitching scheme.
Objective functionalities
The interaction of sutures with the arterial tissue may lead to post-surgery complications. The
undesirable conditions can be described in three failure scenarios: suture failure, arterial-wall
tearing, and thrombosis (due to blood leaking) at the anastomotic interface. Suture failure is
caused when the maximum tensile force of the suture, fs, exceeds the suture strength or leads
to slip or relaxation of the knots that bind the stitches together [25]. Note that it is possible
that suture failure may occur due to suture gradual deterioration with time [26]. Arterial-wall
rupture or injury may be caused when the embedding stresses, σs, due to suture-artery contact
interaction (at the stitching holes) exceed the limit value of artery-wall shear strength.
Thrombosis may be caused if the distance between the edges of the two anastomosed artery
segments, xnet, exceeds the typical size of a few red blood cells, leading to internal bleeding.
In order to avoid failure altogether, the following objective functionalities must be satisfied:
(f )
ultimate shear strength of arterial tissue (σ
max f s < ultimate axial strength of suture/knot
max σ s <
s ,u
max xnet < 3 x red blood cell diameter ( 3d rbc )
(10)
s ,u
/ 2)
(11)
(12)
In addition to the above objective functionalities, the following geometric constraint must be
satisfied to assure adequate stitching spacing:
π R ≥ 4 N s rs
(13)
It should be noted that mechanical changes to the arterial walls and sutures may occur over a
time span of several weeks after surgery. In particular, the wall thickness of the sutured artery
may decrease with time, as is the case of the inflammatory response after surgery. Moreover,
the elastic properties and strength of the artery may change with time due to chemical change
of the suture and its interaction with the arteries [26]. Such long-term implications lead to
lower values of the elastic and strength properties of the arterial walls and suture materials.
Suture-artery interaction
On account of the fact that blood is an incompressible fluid, the radial and longitudinal modes
of arterial response are coupled. Under the applied blood pressure, the artery distends radially
by u(t), and, in order for the blood volume to be maintained, its axial length is decreased from
L to l, resulting in the formation of a gap xg (Figure 4(b)). Conservation of the blood volume
means that the cylindrical volume V(a) = V(b) (Figures 4(a,b)). Then, the decreased
anastomosis length at any time t is given by
l (t ) =
LR 2
( R + u (t ) )
2
(14)
The separation distance l(t) given by equation (14) implies the following solid–fluid
interaction procedure: (a) the blood volume fills the two parts of the anastomosis after
completing the stitching, and (b) pressure is applied leading to contraction along the length of
the initially emptied artery.
The gap developed in the unrestrained (without sutures) state of the artery (Figure 4(b)), is
then determined as the difference between the initial length of the artery (2L) and the length
of the unrestrained deformed state (2l):


R2
xg (t ) = 2 L 1 −
2
 ( R + u (t ) ) 
(15)
Therefore, the resulting net gap developed in the restrained (with sutures) anastomotic region
(Figure 4(c)) can be derived from
xnet (t ) = xg (t ) − 2∆l (t )
(16)
where ∆l is the tensile deformation due to the stitching stiffness.
The tensile forces developed in the suture and arterial tissue (Figure 4(d)) are given
respectively by
f s (t ) = As Esε s (t ) + f s0 =
As Es
xnet (t ) + f s0
ls
FL (t ) = EL AL (t )ε L (t ) = 2π hEL ( R + u (t ) )
(17)
∆l (t )
l (t )
(18)
where εs is the suture strain, εL is the strain of one artery segment, and AL is the crosssectional area of the artery.
The tensile deformation ∆l can be derived from equilibrium of forces in the axial direction,
FL(t) = aNsfs(t), yielding (see Appendix A4)


 f 0 
R2
aN s LR 2  As Es L 1 −
+ s ls 
2
 ( R + u (t ) )  2 

∆l (t ) =
3
π ELls h ( R + u (t ) ) + aN s As Es LR 2
(19)
Substituting equations (15) and (19) into equation (16), we obtain the net gap between the
anastomosed artery segments as
 2π LE l h ( R + u (t ) ) ( R + u (t ) )2 − R 2  − f 0 aN LR 2l
L s
s
s

 s

, for FL (t ) > aN s f s0
3
2
(20)
xnet (t ) = 
π EL ls h ( R + u (t ) ) + aN s As Es LR

, for FL (t ) ≤ aN s f s0
0
Note that a gap across the anastomotic interface will be formed only if the tension developed
in the arterial tissue exceeds the total suture pre-tension.
Upon calculating the anastomotic gap xnet, the suture tensile force fs developed in each stitch
can be readily obtained from equation (17). In addition, embedding stresses σs are developed
because of suture-artery contact interaction at the stitching holes. The embedding stress
induced on the arterial wall is approximated [27] by
σ s (t ) =
af s (t )
2rs h
(21)
It is worth noting that, although based on a linear-elastic model, the system response depends
on a considerable number of parameters. In particular, the solution contains as many as
seventeen input parameters (L, R, Ns, h, Eθ, EL, ps, pd, ts, tcp, ρ, u0, ls, Es, rs, a,
) related to
the geometric and mechanical properties of sutures and arterial walls, the number of sutures,
the loading characteristics, the longitudinal residual stresses, and suture pre-tensioning.
For completeness, the general solution of an artery/graft end-to-end anastomosis is presented
in Appendix B.
Results and discussion
The three response quantities of interest, the anastomotic gap xnet, the suture tensile force fs,
and the embedding stress σs, are directly connected to the aforementioned failure modes. On
normalizing by 2 L, ELh2 and EL respectively, equations (20), (17) and (21) become
  u (t )   u (t )  2 
af s0
r
+
−
−
1
1
Ns s
 1 +

 

R  
R 
R
 2π EL rs h

, for FL (t ) > aN s f s0
xnet (t ) 
3
=
Es
A L
 u (t ) 
2L
Ns s
1 +
 +a

R 
EL
π Rh ls


0
, for FL (t ) ≤ aN s f s0
(22)
2

 af s0 ls  u (t ) 3
Es rs  u (t )   u (t ) 
 2π a
1 +
  1 +
 − 1 +
1 +

EL h 
R  
R 
R 

 EL rs h L 
, for FL (t ) > aN s f s0

3
 u (t )  h ls
f s (t ) 
Es
rs 
a
1
a
N
+
+
=





s
R  rs L
EL
R 
EL h 2 


0
 af s   rs 
, for FL (t ) ≤ aN s f s0




 EL hrs   ah 
(23)
2
3


Es  u (t )   u (t ) 
af s0 h ls  u (t ) 
π
a
1
+
1
+
−
1
+
1
+



 



R  
R 
R 
 2 EL rs h rs L 
 EL 
, for FL (t ) > aN s f s0
3
σ s (t ) 
E
r
 u (t )  h ls
(24)
=
1+
+ a s Ns s


EL
R
r
L
E
R



s
L
 af 0
s

, for FL (t ) ≤ aN s f s0
 2 EL rs h
From equations (22), (23), (24), we observe that the seventeen input parameters of the
problem can be reduced into five dimensionless parameters, namely aEs/EL, L/ls, Nsrs/R, rs/h,
/
. The normalized response quantities for parameter values varied within the
physiological range are presented graphically in Figures 5, 6, 7, 8 and 9.
Figure 5 Normalized anastomotic gap versus normalized radial displacement. Plots for
E
A
different values of product P4 = a s N s s L and for f s0 = 0 .
EL
π Rh ls
Figure 6 Normalized anastomotic gap versus normalized radial displacement. (a) Plots
for different values of parameter aEs/EL and for Ns = 20, As/hR = 0.05, 2L/ls = 100, f s0 = 0 ,
(b) Plots for different values of parameter aEs/EL and for Ns = 12, As/hR = 0.005, 2L/ls = 50,
f s0 = 0 , (c) Plots for different values of parameter 2L/ls and for Ns = 20, As/hR = 0.05, aEs/EL
= 3000, f s0 = 0 , (d) Plots for different values of parameter 2L/ls and for Ns = 12, As/hR =
0.005, aEs/EL = 1000, f s0 = 0 .
Figure 7 Normalized anastomotic gap versus normalized radial displacement. (a) Plots
for different values of parameter As/hR and for Ns = 20, aEs/EL = 3000, 2L/ls = 100, f s0 = 0 ,
(b) Plots for different values of parameter As/hR and for Ns = 12, aEs/EL = 1000, 2L/ls = 50,
f s0 = 0 , (c) Plots for different values of parameter Ns and for aEs/EL = 3000, As/hR = 0.05,
2L/ls = 70, f s0 = 0 , (d) Plots for different values of parameter Ns and for aEs/EL = 1000, As/hR
= 0.005, 2L/ls = 50, f s0 = 0 .
Figure 8 Normalized tensile force in each stitch versus normalized radial displacement.
(a) Plots for different values of parameters 2L/ls and aEs/EL, (b) Plots for different values of
parameters 2L/ls and Ns. ( f s0 = 0 ).
Figure 9 Normalized embedding stress versus normalized radial displacement. Plots for
different values of parameter aEs/EL, Ns and for f s0 = 0 .
The normalized anastomotic gap xnet/2L depends on the product of four dimensionless
parameters, namely aEs/EL, L/ls, Nsrs/R, rs/h, abbreviated herein as P4, and suture pre-tension
parameter af s0 / rs hEL . However, utilizing plausible range of parameter values, we observe
that the contribution of rs/h is relatively small. Figure 5 plots the normalized gap as a function
of the normalized radial displacement u/R, for different values of the product P4, assuming
zero suture pre-tension.
Although Figure 5 presents the variation of the anastomotic gap in a mathematically complete
way, we opted to provide this information in Figures 6 and 7 in a more elaborate and
practically appealing manner, in terms of the design parameters aEs/EL, L/ls, As/hR, Ns, in
order to provide simpler and useful graphs for the optimal selection of materials and
improved functionality of sutures. In particular, Figures 6 and 7 highlight the influence of the
variation of the suture stiffness (Figures 6(a,b)), the stitch length (Figures 6(c,d)), the suture
cross-section area (Figures 7(a,b)), and the number of stitches (Figures 7(c,d)) on the
anastomotic gap, for two different sets of parameters. The results suggest that increasing the
value of any of the design parameters yields a decreased anastomotic gap. In particular, the
most influential parameter in drastically reducing the anastomotic gap is the number of
utilized stitches, Ns, as can be seen from Figures 7(c,d).
Figure 8 plots the normalized tensile force in each stitch (suture strain) as a function of the
normalized radial displacement for different values of parameters 2L/ls, aEs/EL and Ns
(assuming f s0 = 0 ). It can be observed that the normalized suture tensile force is decreased
as the number of stitches is increased, whereas the ratio of suture-to-artery elastic modulus
and the normalized stitch length do not affect significantly the tensile force developed in each
stitch. The latter is also true for the suture radius as suggested by equation (23).
Figure 9 plots the normalized embedding stress due to suture-artery contact interaction as a
function of the normalized radial displacement for different values of parameters aEs/EL and
Ns. It can be seen from Figure 9 that in order to reduce the embedding stress, the number of
stitches must be increased, whereas the parameter aEs/EL plays an insignificant role.
Moreover, the embedding stress becomes smaller with increasing suture radius, as can be
seen from equation (24).
It should be noted that, for a typical anastomosis scheme (with parameters within the
physiological range) and for FL (t ) ≤ aN s f s0 , when the value of pre-tension f s0 exceeds a
certain value (derived from af s0 / 2rs h > σ s.u / 2 ) the arterial wall is likely to fail. On the other
hand, for lower values of pre-tension and for FL (t ) > aN s f s0 , the application of suture pretension can result in reducing the anastomotic gap (equation (22)), while not affecting
considerably the embedding stress (equation (23)) and suture tensile force (equation (24)).
Design considerations
For design purposes the peak values of response are considered. For the general case where
FL (t ) > aN s f s0 , the failure scenarios discussed previously can be prevented by recasting the
inequalities (10), (11), (12) in the form:
Ns >
π EL h( R + umax ) 
af s ,u LR 2
2
2
2 L ( R + umax ) − R  −

ls ( f s ,u − f s0 )
As Es
( R + umax )
2

 ≡ N1




af 0 
ls  hσ s ,u − s 


rs 
2π EL ( R + umax )  
2
2

2
Ns >
( R + umax )  ≡ N 2
aL ( R + umax ) − R  −
aσ s ,u rs LR 2  
2π rs Es





Ns >
{
π EL hls ( R + umax ) 2 L ( R + umax ) − R 2  − 3d rbc ( R + umax )
2


2
0
3d rbc aAs Es LR + f s aLR 2ls
2
}≡N
(25)
(26)
(27)
3
where umax stands for the maximum radial displacement of artery, drbc is the red blood cell
diameter (approximately equal to 7 µm), and fs,u, σs,u are known from the suture strength and
the tensile strength of the arterial wall, respectively. The right-hand side of inequalities (25)
to (27) denotes the minimum number of stitches required to prevent suture failure, arterialwall tearing, and development of excessive gap, respectively. Obviously, the final selection
will be the maximum of N1, N2, N3. However, the number of stitches should not violate the
geometric constraint of equation (13), which practically can be stated as
Ns ≤
πR
4rs
≡ N4
(28)
Therefore, the final selection of number of stitches should be bounded by
max { N1 , N 2 , N 3 } < N s ≤ N 4
(29)
Failure to satisfy inequality (29) means that the material selection and geometric parameter
must be rethought. Typical values related to suture materials indicate that N1 < N2 or N3,
although deteriorated stitches as well as the presence of sutures knots can change this.
When the suture strength is larger than the knot strength, the stitches will fail on the knot
region, otherwise the failure will occur elsewhere. Experiments on the mechanical properties
of different suture materials were performed by Brouwers et al. [25]. Table 1 reports values
for the tensile strength of plain sutures and the tensile strength range for seven knots under
dynamic loading. Moreover, the arterial longitudinal strength was found to be between 1–3
MPa, based on dynamic biaxial tension tests on human aortic tissues [19].
Table 1 Tensile strength of untied and tied fiber (Based on Brouwers et al. [25])
Suture Material
Diameter (mm)
Suture strength (N)
Knot strength (N)
Plain catgut
0.36
25.5
23.7 - 29.6
Maxon
0.31
34.5
22.1 - 46.1
PDS
0.3
27.2
12.4 - 36.5
Prolene
0.26
16.7
6.2 - 26.7
Dexon
0.24
29.1
24.1 - 39.4
Mersilene
0.26
28.3
37.8 - 20.5
Vicryl
0.29
34.6
14.1 - 38.8
Note that, for the particular case where FL (t ) ≤ aN s f s0 , the derived inequalities (25) to (27)
are not valid. In this case, the potential failure is not dependent on the number of sutures Ns,
but rather on whether the pre-tension exceeds either the suture strength or artery strength.
Numerical example
To illustrate the applicability of the proposed analytical model, a design example is
presented, in which the minimum number of stitches required to prevent suture failure,
arterial-wall tearing, and development of excessive anastomotic gap, is calculated for a given
set of typical artery and suture parameters (Table 2). More values for the mechanical
properties of human ascending thoracic aorta can be found in the study of Gozna et al. [28].
Table 2 Parameters used in numerical example
Parameter
Artery
Length, L (cm)
Radius, R (cm)
Thickness, h (cm)
Arterial tissue density, ρ (kg m−3)
Initial displacement, u0 = (2/3)ust (mm)
Circumferential Young’s modulus, Eθ (kPa)
Longitudinal Young’s modulus, EL (kPa)
Tissue strength, σs,u (MPa)
Red blood cell diameter, drbc (µm)
Suturing (Continuous, Prolene)
Length, ls (cm)
Radius, rs (mm)
Young’s modulus, Es (GPa)
Participation factor, a
Suture pre-tension, f s0 = 0 (N)
Suture strength, fs,u (N)
Value
3
0.6
0.11
1160
0.499
700
400
3
7
0.2
0.13
1.5
1.7
0
16.7
Loading
Systolic pressure, ps (mmHg)
Diastolic pressure, pd (mmHg)
Systolic duration, ts (sec)
Cardiac pulse duration, tcp (sec)
120
80
0.35
1
Based on these parameter values, the maximum arterial response, occurring during the
systolic phase, is calculated as umax = 0.997 mm. The maximum circumferential strain εθ,max =
umax/R = 16.6 % is within the validity range of the small-deformation assumption. Based on
inequality (29), the optimal selection of the number of stitches for this example is Ns = 17.
For the selected value of the design parameter Ns, the response quantities of interest are
derived as: suture force fs = 0.24 N (< 16.7 N ≡ fs,u), embedding stress σs = 1.43 MPa (< 1.5
MPa ≡ σs,u/2), and anastomotic gap xnet = 6.02 µm (< 21 µm ≡ 3drbc). As expected, by virtue
of satisfying simultaneously the objective functionalities given by equations (10), (11), (12),
all response quantities fall within the accepted range of values, preventing any of the
aforementioned failure scenarios. Nevertheless, the calculated embedding stress is marginally
acceptable, and the slightest increase of its value may lead to arterial-wall tearing. That is,
despite the fact that the suture can withstand tensile force up to 16.7 N, any suture pre-tension
f s0 > hrsσ s ,u / a = 0.24 N applied by the surgeon in tying the knot may cause arterial injury.
Validation of the model
The present model is fully analytic and has been conceived to be simple with minimum
computational costs, and hence suitable for potential clinical application. The model
incorporates a plethora of the most important-to-the-surgeon parameters for the first time, at
the expense however of strong simplifying hypotheses. One main simplification is the
linearization of the mechanical response of the anastomosis walls. Moreover, anisotropy in
the circumferential and longitudinal direction has been retained also in an approximate way,
ignoring Poisson effects. In addition, failure criteria based on octahedral equivalent stresses
may not be completely appropriate for describing the strength of the arterial tissue. Finally, a
limit-state analysis has been adopted for the failure mechanism of arterial tissues subject to
the loading condition imposed by the stitches.
The aforementioned issues, important by themselves, do not change the holistic view of the
present paper. The linearization of all presented responses gives consistent strains of the order
of 20%. The use of more elaborate hyperelastic constitutive laws does not change appreciably
the central results of our work. Linear-elasticity estimates can be adjusted by appropriate
changes of the elastic moduli. Poisson effects can reduce the stitching results by about 30%,
thus ignoring the Poisson effects is not against safety. Failure of the arterial walls is still an
uncharted topic. It is most probable that failure depends on energy criteria, and in this respect
the shear stress used in this work corresponds to a critical deviatoric energy. Finally, the
limit-state analysis based on a critical shear stress can be easily recast into a tearing criterion
based on the almost-uniaxial state of stress on the sides of the stitches (the linear-elasticity
local model predicts a stress concentration factor of about two).
Although the literature contains several experimental studies dealing with the compliance of
the anastomotic region [9-13], we found that many parameters that seem to affect the suture
stressing are not reported (e.g. the number of stitches Ns). Our present work indicates that
more details regarding the suture material and suturing technique should be reported,
especially if the para-anastomotic hypercompliant zone (PHZ) phenomenon needs to be
addressed. Previous experimental studies of end-to-end anastomosis between isocompliant
arteries or grafts investigate the compliance of the anastomotic region, whereas the main
response quantities calculated in this study (xnet, fs, σs) are not reported in experimental
studies. Nevertheless, it is shown that the present study provides a good estimation of the
compliance value of the anastomotic region with respect to the published experimental
results.
Compliance (C) is the circumferential strain of the systolic phase in respect to the strain of
the diastolic phase εsd divided by the pressure difference:
C=
Ds − Dd
ε sd
=
Dd ( ps − pd ) ( ps − pd )
(30)
where Ds and Dd are the arterial diameters under systolic and diastolic pressure, respectively.
Hasson et al. [10,11] calculated the compliance of dog arterial grafts under dynamic loading.
The compliance away from the PHZ was 0.06% mmHg−1 for the first study and 0.05%
mmHg−1 for the later study. Ulrich et al. [13] calculated the compliance of pig arterial grafts
under dynamic loading as 0.075% mmHg−1. The calculated compliance of our numerical
example is 0.12% mmHg−1. Given that the mechanical data and pressure profile data were not
available for most of the experimental studies and that our model is subjected to pulse
loading of the first loading cycle (meaning that the calculated displacements may be up to
two times larger than the static or long-term dynamic loading), our model constitutes a good
approximation of the experimental results.
Of particular interest is the PHZ phenomenon. Hasson et al. [11] found that the PHZ
phenomenon occurs more frequently for anastomosis of the continuous stitching technique
than the interrupted stitching technique. Figure 10(a) shows the schematic compliance along
the anastomotic region. The PHZ phenomenon (region 2) is pronounced in the case of
continuous stitches, whereas away from the anastomosis zone the compliance is constant
(region 1). From our study, the net gap xnet is increased by 15% in the case of continuous
stitching compared to the case of interrupted stitching. This may justify the decreased
longitudinal stretch ∆l/L and lower tangent elastic modulus Eθ1 > Eθ2 (Figure 10(b)) of the
continuous stitching case. The decrease of tangent elastic modulus results to higher
compliance at the PHZ.
Figure 10 Schematic correlation of PHZ phenomenon to the stiffness of the arterial
tissue. (a) Compliance of the anastomotic region. The PHZ phenomenon (region 2) exists in
the case of continuous stitches, as was concluded by Hasson et al. [11]. Away from the
anastomosis zone the compliance is constant (region 1), (b) Circumferential stresslongitudinal strain relationship of a nonlinear hyperelastic material. In the case of continuous
stitching, the longitudinal stretch is lower than that of the discrete stitching. This results to a
lower tangential modulus (Eθ1 > Eθ2), implying a higher compliance at the PHZ.
The experimental results suggest a decrease of stiffness by about 29% [11]. From the
numerical example presented in our study, the total longitudinal stretch away from the suture
line is 1.36. The longitudinal stretch at the PHZ is reduced by 29% compared to the
longitudinal stretch away from the anastomotic region. Taking a nonlinear constitutive law
according to Skalak et al. [29], the continuous stitching (stretch 1.27) decreases the tangent
modulus by 24% in comparison to the interrupted stitching (stretch 1.36), indicating that the
increase of compliance at the PHZ can be correlated to the decrease of stiffness, as Hasson et
al. [11] suggest.
In conclusion, the present model, even though simple and approximate, captures adequately
the essence of the phenomenon. More complex models can be important in refining the
present results, but on the other hand will require more material data that may be difficult to
obtain or assess their direct contribution.
Conclusions
Presented in this study is the mathematical formulation governing the dynamic behavior of
end-to-end arterial anastomosis, with emphasis on suture-artery interaction and the axialcircumferential deformation coupling in the artery response. Closed-form, time-dependent
expressions were derived for the system response, in terms of the radial displacement of
artery (equation (4)), the anastomotic gap (equation (20)), the suture tensile force (equation
(17)), and the embedding stress due to suture-artery contact interaction (equation (21)). It is
worth noting that, although linear elastic, the model is comprehensive in that it captures the
effects of all pertinent parameters, including the geometric and mechanical properties of
sutures and arterial walls, the number of sutures, the loading characteristics, the longitudinal
residual stresses, and suture pre-tensioning. As a result, the response was obtained as a
function of as many as seventeen input parameters (L, R, Ns, h, Eθ, EL, ps, pd, ts, tcp, ρ, u0, ls,
Es, rs, a,
). Nevertheless, on normalizing appropriately the response quantities, the
problem can be described by only five dimensionless parameters (aEs/EL, L/ls, Nsrs/R, rs/h,
af s0 / rs hEL ).
Inherent in the analysis are limitations stemming from the underlying model assumptions (as
discussed in section Methods). We are currently working on similar problems where we
gradually relax assumptions made regarding the artery cylindrical geometry, the material
linear constitutive relations, the arterial-wall homogeneity, and the kinematic conditions.
Findings obtained by the suture-tissue interaction analysis reveal the nonlinear dependency of
the system response on the radial extension of artery and highlight useful interrelations
among the problem parameters. In regard to the normalized anastomotic gap, the results
suggest that increasing the value of any of the design parameters, excluding
, yields a
decreased anastomotic gap. In particular, the most influential parameter in drastically
reducing the anastomotic gap is the number of utilized stitches, Ns, as can be seen from
Figures 7(c,d). The normalized suture tensile force is instead affected only by the number of
stitches. A higher number of utilized stitches results in a smaller tensile force developed in
each stitch (Figure 8). It has also been shown that the normalized embedding stress is
decreased as the number of stitches is increased, whereas the influence of the ratio of sutureto-artery elastic modulus on the embedding stress is insignificant (Figure 9).
It should be noted that, for a typical anastomosis scheme (with parameters within the
physiological range) and for, when the value of pre-tension f s0 exceeds a certain value
(derived from af s0 / 2rs h > σ s.u / 2 ) the arterial wall is likely to fail. On the other hand, for
lower values of pre-tension and for FL (t ) > aN s f s0 , the application of suture pre-tension can
result in reducing the anastomotic gap (equation (22)), while not affecting considerably the
embedding stress (equation (23)) and suture tensile force (equation (24)).
In conclusion, the primary contribution of this study is the development of a fundamental
analytical model that predicts the dynamic behavior of end-to-end arterial anastomosis.
Derived from first principles, thus characterized by generality, the proposed model offers new
and better-substantiated understanding of the mechanics of end-to-end anastomosis scheme.
The mathematical formulation reveals useful interrelations among the problem parameters,
thus making the proposed model a valuable tool for the optimal selection of materials and
improved functionality of sutures. The comprehensive failure criteria established in this study
can ultimately form the basis for the development of vascular anastomosis guidelines
pertaining to the prevention of post-surgery implications.
Appendix A
A1: proof of equation (1)
By considering the equilibrium of the element of the circular ring with unit length shown in
Figure 1 we obtain
 dθ
p(t ) Rdθ − N (t ) sin 
 2

 dθ
 − N (t )sin 

 2
d 2u (t )

=
m

dt 2

(a.1)
where m = ρRhdθ is the mass of the arterial element of unit length. From Hooke's law, the
axial force N is given by
N (t ) = Eθ h
u (t )
R
(a.2)
On substituting the above expression in equation (a.1), and by considering small angles (so
that sin(dθ/2) ≈ dθ/2), we obtain
p(t ) Rdθ − Eθ h
u (t )
d 2u (t )
dθ = ρ Rhdθ
R
dt 2
(a.3)
By dividing by Rdθ we obtain the equation governing the radial displacement response as
p(t ) − Eθ h
u (t )
d 2u (t )
=
ρ
h
R2
dt 2
(a.4)
A2: proof of equation (4)
The total response of the system to a pulse loading with nonzero initial conditions is the sum
of the response to the pulse loading up(t) and the response to free vibration uf(t) due to initial
conditions.
The response to free vibration with initial displacement uf(0) = u0 and initial velocity
u f (0) = 0 is given by
u f (t ) = u0 cos ωnt ,
0 ≤ t ≤ tcp
(a.5)
The response of the system to impulse force p(t) can be determined by using the convolution
(Duhamel) integral. A convolution integral is simply the integral of the product of the
external force p(τ) and the unit-impulse response function of the system h(t − τ):
t
1
u p (t ) = ∫ p (τ )h(t − τ )dτ =
mωn
0
t
∫ p(τ ) sin [ω (t − τ )]dτ
(a.6)
n
0
The response in the systolic phase (0 ≤ t ≤ ts), in which the system is subjected to constant
force p(τ) = ps, is calculated as
u p I (t ) =
1
mωn
t
∫ ps sin [ωn (t − τ )]dτ =
0
ps R 2
(1 − cos ωnt ) ,
Eθ h
t ≤ ts
(a.7)
The response in the diastolic phase (ts ≤ t ≤ tcp), in which the system is subjected to impulse
force p(τ) = ps − (ps − pd)(τ − ts)/( tcp − ts), is derived from
u p II (t ) =
1
mωn
t

∫  p
ts

s
−

ps − pd
(τ − ts )  sin [ωn (t − τ )]dτ + u p I (t s ) cos ωn (t − t s )
tcp − ts

+
u p I (t s )
ωn
sin ωn (t − ts ),
(a.8)
t s <t ≤ tcp
in which the first term concerns the force-vibration response associated with the diastolic
phase loading, and the last two terms concern the free-vibration response due to initial
I
conditions u I(t ) and u p (t s ) induced at the end of the systolic phase. On carrying out the
p
s
calculations, equation (a.8) simplifies to
u p II (t ) =
ps R 2
Eθ h
 ps − pd

 tcp − ts


sin ωn ( t − ts ) 
t s − t +
 + ps (1 − cos ωnt )  , t s < t ≤ tcp
ωn



(a.9)
The total displacement is then obtained as the sum of the pulse-loading response up(t) and the
free-vibration response uf(t):

ps R 2
u
cos
ω
t
+
(1 − cos ωnt ) ,
n
 0
E
h
θ


u (t ) = 

2 

sin ωn ( t − t s ) 

 u0 cos ωn t + ps R  ps − pd t s − t +
 + ps (1 − cos ωn t )  ,
ωn
Eθ h  tcp − t s 



0 ≤ t ≤ ts
(a.10)
ts < t ≤ tcp
A3: proof of equations (6), (7), (8), (9)
The maximum displacement of the arterial system may occur either during the systolic phase
(0 ≤ t ≤ ts) or during the diastolic phase (ts < t ≤ tcp). The maximum displacement of the
systolic phase (for |u0|/ust < 1) occurs for cos ωnt = − 1. Therefore, by substituting this
expression into the first part of equation (4) the maximum displacement of the systolic phase
I
is obtained as
umax
I
umax
=
2 ps R 2
− u0
Eθ h
(a.11)
To calculate the time instant t1 corresponding to the maximum response of the diastolic
phase, the derivative of the displacement with respect to time is set equal to zero:
ps R 2  ps − pd
du (t )
=

ωn Eθ h  tcp − t s
dt

 ps R 2 R 2 ps − pd sin ωn ts

+
− u0 
 + sin ωn t1 
ωn
 Eθ h Eθ h tcp − ts


 R 2 ps − pd sin ωn ts 
+ cos ωn t1 
=0
ωn 
 Eθ h tcp − ts
(a.12)
which can be recast in the following form

B 
B1 = B2 2 + B32 cos  ωnt1 − tan −1 2 
B3 

(a.13)
where
ps R 2  ps − pd
B1 = −

ωn Eθ h  tcp − ts



ps R 2 R 2 ps − pd sin ωn ts
B2 =
+
− u0
Eθ h Eθ h tcp − t s
ωn
B3 =
R 2 ps − pd sin ωn t s
Eθ h tcp − t s
ωn
(a.14)
(a.15)
(a.16)
On solving for t1 we get the time instant corresponding to the maximum response of the
diastolic phase as
t1 =
1  −1
B1
B  1
 cos
+ tan −1 2  =
cos −1 A1 + tan −1 A2 )
(
2
2
ωn 
B3  ωn
B2 + B3


(a.17)
II
The maximum displacement of the diastolic phase umax is calculated at t = t1 as
II
umax
= u0 cos ωnt1 +
R 2  ps − pd

Eθ h  tcp − t s


sin ωn ( t1 − t s ) 
 t s − t1 +
 + ps (1 − cos ωnt1 ) 
ωn



(a.18)
The overall maximum response is then obtained as
umax
 2 ps R 2

− u0 , u0 cos ωn t1 +


Eθ h


I
II
, umax
= max {umax
} = max  2  p − p 
 (a.19)

sin ωn ( t1 − ts ) 
R
s
d
t −t +
 + ps (1 − cos ωn t1 )  
 Eθ h  tcp − ts  s 1
ω
 
n




A4: proof of equation (19)
The equilibrium of forces in the axial direction requires that
FL (t ) = aN s f s (t )
(a.20)
Substituting equations (17) and (18) into equation (a.20), the equilibrium of forces in the
axial direction yields
2π hEL ( R + u (t ) )
AE

∆l (t )
= aN s  s s xnet (t ) + f s0 
l (t )
 ls

(a.21)
By combining equations (15) and (16), the net gap between the anastomosed artery segments
is derived as


R2
xnet (t ) = 2 L 1 −
− 2∆l (t )
2 
 ( R + u (t ) ) 
(a.22)
Substituting equation (a.22) into equation (a.21), the equilibrium equation is expressed in
terms of ∆l(t) as
2π hEL ( R + u (t ) )
 A E




∆l (t )
R2
0

= aN s  s s 2 L  1 −

−
2
∆
l
(
t
)
+
f
s 
2
l (t )
 ( R + u (t ) ) 

 ls

(a.23)
which can be readily solved for the tensile deformation:


 f 0 
R2
aN s LR 2  As Es L 1 −
+ s ls 
2 
 ( R + u (t ) )  2 

∆l (t ) =
3
π ELls h ( R + u (t ) ) + aN s As Es LR 2
(a.24)
Appendix B: Solution of end-to-end anastomosis between
artery and graft material
This section presents the general solution of an end-to-end anastomosis between a host artery
and a graft, each one having different geometrical and mechanical properties. The artery
segment has length La, radius Ra, thickness ha, and Young’s modulus in the longitudinal
direction and circumferential direction ELa and Eθa, respectively, whereas the graft has length
Lg, radius Rg, thickness hg, and Young’s modulus in the longitudinal direction and
circumferential direction ELg and Eθg, respectively (Figure 11(a)). The conservation of the
blood volume requires that the artery initial length La decrease to la and the graft initial length
Lg decrease to lg (Figure 11(b)) according to:
Figure 11 Artery-graft end-to-end anastomosis analysis. (a) Anastomosis model (at-rest
state); the artery and graft are clamped at the far ends and no pressure is transmitted at this
stage since the artery is emptied from the blood, (b) unrestrained deformed state (without
sutures); the blood volume is conserved, (c) deformed state of anastomotic region due to
dynamic loading, (d) forces acting on end-element of artery segment, (e) forces acting on
end-element of graft segment.
La Ra 2
la =
( Ra + ua (t ))2
(b.1)
lg =
Lg Rg 2
(b.2)
( Rg + u g (t )) 2
where ua and ug are the radial deformations of the artery and graft, respectively. Note that the
graft has not initial radial displacement due to residual stresses. The gap developed in the
unrestrained (without sutures) state of the artery is determined as
Lg Rg 2
La Ra 2
xg (t ) = La + Lg − la (t ) − l g (t ) = La + Lg −
−
( Ra + ua (t ))2 ( Rg + u g (t ))2
(b.3)
Therefore, the resulting net gap developed in the restrained (with sutures) anastomotic region
can be derived from
xnet (t ) = xg (t ) − ∆la (t ) − ∆lg (t )
(b.4)
where ∆la is the tensile deformation due to the artery/stitches interaction, and ∆lg is the tensile
deformation due to the graft/stitches interaction (Figure 11(c)).
The tensile forces developed in the suture, arterial tissue, and graft are given respectively by
f s (t ) = As Esε s (t ) + f s0 =
As Es
xnet (t ) + f s0
ls
(b.5)
∆la (t )
la (t )
(b.6)
FLa (t ) = 2π ha ELa ( Ra + ua (t ) )
FL g (t ) = 2π hg ELg ( Rg + u g (t ) )
∆lg (t )
lg (t )
(b.7)
The unknown tensile deformations ∆la and ∆lg can be derived from equilibrium of forces in
the axial direction, FLa(t) = FLg(t) and FLa(t) = aNsfs(t) (Figure 11(d,e)), yielding
 



Rg 2
Ra 2


aN s f + aN s As Es  La 1 −
+ Lg 1 −
2 
2 
 ( R + u (t ) )  
  ( Ra + ua (t ) ) 
g
g


∆la (t ) =
3 
2

aN s As Es  ha ELa Lg Rg ( Ra + ua (t ) ) 
2π ELa ha
3
1+
( Ra + ua (t ) ) +
La Ra 2
ls
 h E L R 2 ( R + u (t ) )3 
g Lg a a
g
g


0
s
∆lg (t ) =
ha ELa Lg Rg 2 ( Ra + ua (t ) )
hg ELg La Ra
2
(R
g
(b.8)
3
+ u g (t ) )
3
∆la (t )
(b.9)
Substituting equations (b.3), (b.8) and (b.9) into equation (b.4), we obtain the net gap
between the anastomosed artery segments as
 



Rg 2
Ra 2



2π ELa ha ( Ra + ua (t ) )  La 1 −
+ Lg 1 −
2 
2 
 ( Rg + ug (t ) )  
  ( Ra + ua (t ) ) 


xnet (t ) =
3 
2

aN s As Es La Ra 2  ha ELa Lg Rg ( Ra + ua (t ) ) 
3
2π ELa ha ( Ra + ua (t ) ) +
1+
ls
 hg ELg La Ra 2 ( Rg + u g (t ) )3 


 h E L R 2 ( R + u (t ) )3 
a
a
0
2 

aN s f s La Ra 1 + a La g g
2
 hg ELg La Ra ( Rg + u g (t ) )3 


−
3 
2

aN s As Es La Ra 2  ha ELa Lg Rg ( Ra + ua (t ) ) 
3
2π ELa ha ( Ra + ua (t ) ) +
1+
ls
 hg ELg La Ra 2 ( Rg + u g (t ) )3 


3
(b.10)
Note that a gap across the anastomotic interface will be formed only if the tension developed
in the arterial tissue exceeds the total suture pre-tension. The suture tensile force fs developed
in each stitch can be obtained from equation (17). The embedding stresses induced on the
arterial wall σsa and graft wall σsg must be compared to the strength of the artery σsa,u and
strength of the graft σsa,u, respectively:
σ sa (t ) =
af s (t )
< σ sa ,u / 2
2rs ha
(b.11)
σ sg (t ) =
af s (t )
< σ sg ,u / 2
2rs hg
(b.12)
Competing interests
Authors PC Roussis, HP Charalambous, DC Demetriou, and GP Georghiou declare that they
have no conflict of interest. Author AE Giannakopoulos has received research grant under the
“ARISTEIA II” Action of the “OPERATIONAL PROGRAMME EDUCATION AND
LIFELONG LEARNING”, co-founded by the European Social Fund (ESF) and National
Resources.
Authors’ contributions
PCR and AEG conceived the idea of the study, designed and coordinated the work, identified
the model parameters, and formulated the mathematical model. HPC and DCD contributed to
the model development, carried out the numerical computations, and prepared the first draft
of the manuscript. PCR and AEG revised and critically reviewed the final manuscript
submitted for publication. PCR, AEG and HPC interpreted and discussed the derived results
and findings of the study. GPG contributed discussions and suggestions throughout this
project, and provided pertinent data. All authors read and approved the final manuscript.
Acknowledgments
AE Giannakopoulos acknowledges that part of this project was implemented under the
“ARISTEIA II” Action of the “OPERATIONAL PROGRAMME EDUCATION AND
LIFELONG LEARNING” and is co-founded by the European Social Fund (ESF) and
National Resources.
References
1. Ballyk PD, Walsh C, Butany J, Ojha M. Compliance mismatch may promote graft-artery
intimal hyperplasia by altering suture-line stresses. J Biomech. 1998;31:229–37.
2. Leuprecht A, Perktold K, Prosi M, Berk T, Trubel W, Schima H. Numerical study of
hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts. J
Biomech. 2002;35:225–36.
3. Perktold K, Leuprecht A, Prosi M, Berk T, Czerny M, Trubel W, et al. Fluid Dynamics,
Wall Mechanics, and Oxygen Transfer in Peripheral Bypass Anastomoses. Ann Biomed Eng.
2002;30:447–60.
4. Sankaranarayanan M, Chua LP, Ghista DN, Tan YS. Computational model of blood flow
in the aorto-coronary bypass graft. Biomed Eng OnLine. 2005;4:14.
5. Frauenfelder T, Boutsianis E, Schertler T, Husmann L, Leschka S, Poulikakos D, et al.
Flow and wall shear stress in end-to-side and side-to-side anastomosis of venous coronary
artery bypass grafts. Biomed Eng OnLine. 2007;6:35.
6. Cacho F, Doblaré M, Holzapfel GA. A procedure to simulate coronary artery bypass graft
surgery. Med Biol Eng Comput. 2007;45:819–27.
7. Schiller NK, Franz T, Weerasekara NS, Zilla P, Reddy BD. A simple fluid–structure
coupling algorithm for the study of the anastomotic mechanics of vascular grafts. Comput
Methods Biomech Biomed Engin. 2010;13:773–81.
8. Rachev A, Manoach E, Berry J, Moore Jr J. A Model of Stress-induced Geometrical
Remodeling of Vessel Segments Adjacent to Stents and Artery/Graft Anastomoses. J Theor
Biol. 2000;206:429–43.
9. Lyman DJ, Fazzio FJ, Voorhees H, Robinson G, Albo D. Compliance as a factor effecting
the patency of a copolyurethane vascular graft. J Biomed Mater Res. 1978;12:337–45.
10. Hasson JE, Megerman J, Abbott WM. Increased compliance near vascular anastomoses. J
Vasc Surg. 1985;2:419–23.
11. Hasson JE, Megerman J, Abbott WM. Suture technique and para-anastomotic
compliance. J Vasc Surg. 1986;3:591–8.
12. Abbott WM, Megerman J, Hasson JE, L’Italien G, Warnock DF. Effect of compliance
mismatch on vascular graft patency. J Vasc Surg. 1987;5:376–82.
13. Ulrich M, Staalsen N-H, Djurhuus C, Christensen T, Nygaard H, Hasenkam J. In Vivo
Analysis of Dynamic Tensile Stresses at Arterial End-to-end Anastomoses. Influence of
Suture-line and Graft on Anastomotic Biomechanics. Eur J Vasc Endovasc Surg.
1999;18:515–22.
14. Patel DJ, Janicki JS, Carew TE. Static Anisotropic Elastic Properties of the Aorta in
Living Dogs. Circ Res. 1969;25:765–79.
15. Migliavacca F, Dubini G. Computational modeling of vascular anastomoses. Biomech
Model Mechanobiol. 2005;3:235–50.
16. Gozna ER, Marble AE, Shaw A, Holland JG. Age-related changes in the mechanics of the
aorta and pulmonary artery of man. J Appl Physiol. 1974;36:407–11.
17. Holzapfel G, Gasser T, Ogden R. A New Constitutive Framework for Arterial Wall
Mechanics and a Comparative Study of Material Models. J Elast. 2000;61:1–48.
18. Wertheim G. Mémoire sur l’élasticité et la cohésion des principaux tissus du corps
humain. Ann Chim Phys. 1847;21:385–414.
19. Mohan D, Melvin JW. Failure properties of passive human aortic tissue. II—Biaxial
tension tests. J Biomech. 1983;16:31–44.
20. Alastrué V, Peña E, Martínez MÁ, Doblaré M. Assessing the Use of the “Opening Angle
Method” to Enforce Residual Stresses in Patient-Specific Arteries. Ann Biomed Eng.
2007;35:1821–37.
21. Schajer GS, Green SI, Davis AP, Hsiang YN-H. Influence of Elastic Nonlinearity on
Arterial Anastomotic Compliance. J Biomech Eng. 1996;118:445.
22. Zhong L, Ghista DN, Ng EYK, Lim ST, Chua TSJ. Determination of Aortic Pressure–
time Profile, along with Aortic Stiffness and Peripheral Resistance. J Mech Med Biol.
2004;4:499–509.
23. Baumgartner N, Dobrin PB, Morasch M, Dong QS, Mrkvicka R. Influence of suture
technique and suture material selection on the mechanics of end-to-end and end-to-side
anastomoses. J Thor Cardiov Surg. 1996;111:1063–72.
24. Dobrin PB. Polypropylene suture stresses after closure of longitudinal arteriotomy. J Vasc
Surg. 1988;7:423–8.
25. Brouwers J, Oosting D, de Haas D, Klopper PJ. Dynamic loading of surgical knots. Surg
Gynecol Obstet. 1991;173:443–8.
26. Lee S, Fung YC, Matsuda M, Xue H, Schneider D, Han K. The development of
mechanical strength of surgically anastomosed arteries sutured with Dexon. J Biomech.
1985;18:81–9.
27. Salmon CG, Johnson EJ. Steel Structures–Design and Behavior Emphasizing Load and
Resistance Factor Design. 4th ed. New York, NY: Harper Collins College Publishers; 1996.
28. Gozna ER, Marble AE, Shaw AJ, Winter DA. Mechanical properties of the ascending
thoracic aorta of man. Cardiovasc Res. 1973;7:261–5.
29. Skalak R, Tozeren A, Zarda RP, Chien S. Strain Energy Function of Red Blood Cell
Membranes. Biophys J. 1973;13:245–64.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11